Is there an approach to scheme theory in which schemes are viewed as algebraic objects instead of geometric ones?
Here's what I mean: From the functorial perspective, a scheme is a certain kind of functor $F: \text{CRing}\to\text{Sets}$. These functors are geometric objects because they are generalized objects of $\text{CRing}^{\text{op}}$. What if I instead want to consider functors $G: \text{CRing}^{\text{op}}\to \text{Sets}$. These functors would be algebraic objects because they are generalized objects of $\text{CRing}$. Is there a natural way to define the opposite category of schemes in terms of certain functors $G: \text{CRing}^{\text{op}}\to \text{Sets}$? What topology on $\text{CRing}$ could these opposite-schemes be sheaves with respect to?